sigma10-063. 586KB Jun 04 2011 12:10:42 AM
Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 6 (2010), 063, 47 pages
Higher-Dimensional Unif ied Theories
with Fuzzy Extra Dimensions⋆
Athanasios CHATZISTAVRAKIDIS
†‡
and George ZOUPANOS
‡
†
Institute of Nuclear Physics, NCSR Demokritos, GR-15310 Athens, Greece
E-mail: [email protected]
‡
Physics Department, National Technical University of Athens,
GR-15780 Zografou Campus, Athens, Greece
E-mail: [email protected]
Received May 06, 2010, in final form July 22, 2010; Published online August 12, 2010
doi:10.3842/SIGMA.2010.063
Abstract. Theories defined in higher than four dimensions have been used in various
frameworks and have a long and interesting history. Here we review certain attempts,
developed over the last years, towards the construction of unified particle physics models in
the context of higher-dimensional gauge theories with non-commutative extra dimensions.
These ideas have been developed in two complementary ways, namely (i) starting with
a higher-dimensional gauge theory and dimensionally reducing it to four dimensions over
fuzzy internal spaces and (ii) starting with a four-dimensional, renormalizable gauge theory
and dynamically generating fuzzy extra dimensions. We describe the above approaches and
moreover we discuss the inclusion of fermions and the construction of realistic chiral theories
in this context.
Key words: fuzzy extra dimensions; unified gauge theories; symmetry breaking
2010 Mathematics Subject Classification: 70S15
Contents
1 Introduction
2
2 Fuzzy spaces and dimensional reduction
2.1 The fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Gauge theory on the fuzzy sphere . . . . . . . . . . . .
2.2 Dimensional reduction and gauge symmetry enhancement . .
2.3 Non-trivial dimensional reduction over fuzzy extra dimensions
2.3.1 Ordinary CSDR . . . . . . . . . . . . . . . . . . . . .
2.3.2 Fuzzy CSDR . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Solving the CSDR constraints for the fuzzy sphere . .
2.4 The problem of chirality in fuzzy CSDR . . . . . . . . . . . .
3 Dynamical generation of fuzzy extra dimensions
3.1 The four dimensional action . . . . . . . . . . . . . .
3.2 Emergence of extra dimensions and the fuzzy sphere
3.3 Inclusion of fermions . . . . . . . . . . . . . . . . . .
2 . . . . .
3.3.1 Fermions on M 4 × S 2 and M 4 × SN
3.3.2 The spectrum of D
6 (2) in a type I vacuum . .
⋆
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This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is
available at http://www.emis.de/journals/SIGMA/noncommutative.html
2
A. Chatzistavrakidis and G. Zoupanos
3.4
3.3.3 The spectrum of D
6 (2) in a type II vacuum . . . . . .
Dynamical generation of fuzzy S 2 × S 2 and mirror fermions
3.4.1 The action . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Type I vacuum and fuzzy S 2 × S 2 . . . . . . . . . .
2 . . . . . . . . . . . . . . . . .
3.4.3 Operators on SL2 × SR
3.4.4 Type II vacuum and the zero-modes . . . . . . . . .
4 Orbifolds, fuzzy extra dimensions and chiral models
4.1 N = 4 SYM and Z3 orbifolds . . . . . . . . . . . . . .
4.2 Twisted fuzzy spheres . . . . . . . . . . . . . . . . . .
4.3 Dynamical generation of twisted fuzzy spheres . . . .
4.4 Chiral models from the fuzzy orbifold . . . . . . . . .
4.4.1 A SU (4)c × SU (2)L × SU (2)R model . . . . . .
4.4.2 A SU (4)c × SU (4)L × SU (4)R model . . . . . .
4.4.3 A SU (3)c × SU (3)L × SU (3)R model . . . . . .
4.4.4 A closer look at the masses . . . . . . . . . . .
4.5 Fuzzy breaking for SU (3)3 . . . . . . . . . . . . . . . .
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5 Discussion and conclusions
40
A Clif ford algebra conventions
42
References
42
1
Introduction
The unification of the fundamental interactions has always been one of the main goals of theoretical physics. Several approaches have been employed in order to achieve this goal, one of the
most exciting ones being the proposal that extra dimensions may exist in nature. The most
serious support on the existence of extra dimensions came from superstring theories [1], which
at present are the best candidates for a unified description of all fundamental interactions, including gravity and moreover they can be consistently defined only in higher dimensions. Among
superstring theories the heterotic string [2] has always been considered as the most promising
version in the prospect to find contact with low-energy physics studied in accelerators, mainly
due to the presence of the ten-dimensional N = 1 gauge sector. Upon compactification of
the ten-dimensional space-time and subsequent dimensional reduction the initial E8 × E8 gauge
theory can break to phenomenologically interesting Grand Unified Theories (GUTs), where
the Standard Model (SM) could in principle be accommodated [2]. Dimensional reduction of
higher-dimensional gauge theories had been studied few years earlier than the discovery of the
heterotic superstring with pioneer studies the Forgacs–Manton Coset Space Dimensional Reduction (CSDR) [3, 4, 5] and the Scherk–Schwarz group manifold reduction [6]. In these frameworks
gauge-Higgs unification is achieved in higher dimensions, since the four-dimensional gauge and
Higgs fields are simply the surviving components of the gauge fields of a pure gauge theory
defined in higher dimensions. Moreover in the CSDR the addition of fermions in the higherdimensional gauge theory leads naturally to Yukawa couplings in four dimensions. A major
achievement in this direction is the possibility to obtain chiral theories in four dimensions [7].
On the other hand, non-commutative geometry offers another framework aiming to describe
physics at the Planck scale [8, 9]. In the spirit of non-commutative geometry also particle
models with non-commutative gauge theory were explored [10] (see also [11]), [12, 13]. It is
worth stressing the observation that a natural realization of non-commutativity of space appears
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
3
in the string theory context of D-branes in the presence of a constant antisymmetric field [14],
which not only brought together the two approaches but they can be considered complementary.
Another interesting development in the non-commutative framework was the work of Seiberg and
Witten [15], where a map between the non-commutative and commutative gauge theories has
been described. Based on that and related subsequent developments [16, 17] a non-commutative
version of the SM has been constructed [18]. These non-commutative models represent interesting generalizations of the SM and hint at possible new physics. However they do not address
the usual problem of the SM, the presence of a plethora of free parameters mostly related to the
ad hoc introduction of the Higgs and Yukawa sectors in the theory.
According to the above discussion it is natural to investigate higher-dimensional gauge theories and their dimensional reduction in four dimensions. Our aim is to provide an up to-date
overview of certain attempts in this direction, developed over the last years. The development
of these ideas has followed two complementary ways, namely (i) the dimensional reduction
of a higher-dimensional gauge theory over fuzzy internal spaces [19] and (ii) the dynamical
generation of fuzzy extra dimensions within a four-dimensional and renormalizable gauge theory [20, 21, 22, 23].
More specifically, the paper is organized as follows. In Section 2 we present a study of
the CSDR in the non-commutative context which sets the rules for constructing new particle
models that might be phenomenologically interesting. One could study CSDR with the whole
parent space M D being non-commutative or with just non-commutative Minkowski space or noncommutative internal space. We specialize here to this last situation and therefore eventually
we obtain Lorentz covariant theories on commutative Minkowski space. We further specialize
to fuzzy non-commutativity, i.e. to matrix type non-commutativity. Thus, following [19], we
consider non-commutative spaces like those studied in [9, 12, 13] and implementing the CSDR
principle on these spaces we obtain the rules for constructing new particle models. In Section 2.1
the fuzzy sphere is introduced and moreover the gauge theory on the fuzzy sphere is discussed.
In Section 2.2 a trivial dimensional reduction of a higher-dimensional gauge theory over the
fuzzy sphere is performed. In Section 2.3 we discuss the non-trivial dimensional reduction; first
the CSDR scheme in the commutative case is briefly reviewed and subsequently it is applied to
the case of fuzzy extra dimensions. In Section 2.4 the issue of chirality is discussed within the
above context.
In Section 3 we reverse the above approach [20] and examine how a four-dimensional gauge
theory dynamically develops fuzzy extra dimensions. In Sections 3.1 and 3.2 we present a simple
field-theoretical model which realizes the above ideas. It is defined as a renormalizable SU (N )
gauge theory on four-dimensional Minkowski space M 4 , containing three scalars in the adjoint
of SU (N ) that transform as vectors under an additional global SO(3) symmetry with the most
general renormalizable potential. We then show that the model dynamically develops fuzzy
2 . The appropriate interpretation is therefore
extra dimensions, more precisely a fuzzy sphere SN
2 . The low-energy effective action is that of a four-dimensional
as gauge theory on M 4 × SN
4
gauge theory on M , whose gauge group and field content is dynamically determined by com2 . An interesting and rich
pactification and dimensional reduction on the internal sphere SN
pattern of spontaneous symmetry breaking appears, namely the breaking of the original SU (N )
gauge symmetry down to either SU (n) or SU (n1 ) × SU (n2 ) × U (1). The latter case is the
generic one, and implies also a monopole flux induced on the fuzzy sphere. Moreover we
determine explicitly the tower of massive Kaluza–Klein modes corresponding to the effective
geometry, which justifies the interpretation as a compactified higher-dimensional gauge theory.
Last but not least, the model is renormalizable. In Sections 3.3 and 3.4 we explore the dynamical generation of a product of two fuzzy spheres [22]. Specifically, we start with the SU (N )
Yang–Mills theory in four dimensions, coupled to six scalars and four Majorana spinors, i.e.
with the particle spectrum of the N = 4 supersymmetric Yang–Mills theory (SYM). Adding
4
A. Chatzistavrakidis and G. Zoupanos
an explicit R-symmetry-breaking potential, thus breaking the N = 4 supersymmetry, we reveal
2 vacua. In the most interesting case we include magnetic fluxes on the
stable M 4 × SL2 × SR
extra-dimensional fuzzy spheres and study the fermion spectrum, in particular the zero modes
of the Dirac operator. The outcome of our analysis is that we obtain a mirror model in low
energies.
In Section 4 we present a recently developed approach within the above framework, which
leads to chiral low-energy models [23]. In particular, Z3 orbifolds of N = 4 supersymmetric
Yang–Mills theory are discussed and they are subsequently used to dynamically generate fuzzy
extra dimensions. The extra dimensions are described by twisted fuzzy spheres, defined in
Section 4.2. This framework allows to construct low-energy models with interesting unification
groups and a chiral spectrum. In particular, we are led to study three different models based on
the gauge groups SU (4) × SU (2) × SU (2), SU (4)3 and SU (3)3 respectively. The spontaneous
symmetry breaking of the latter unified gauge group down to the minimal supersymmetric
standard model and to the SU (3) × U (1)em is subsequently studied within the same framework.
Finally, Section 5 contains our conclusions.
2
2.1
Fuzzy spaces and dimensional reduction
The fuzzy sphere
The fuzzy sphere [24] is a noncommutative manifold which corresponds to a matrix approximation of the ordinary sphere. In order to describe it let us consider the ordinary sphere as
a submanifold of the three-dimensional Euclidean space R3 . The coordinates of R3 will be denoted as xa , a = 1, 2, 3. Then the algebra of functions on the ordinary sphere S 2 ⊂ R3 can be
generated by the coordinates of R3 modulo the relation
3
X
xa xa = R2 ,
a=1
where R is the radius of the sphere. Clearly, the sphere admits the action of a global SO(3) ∼
SU (2) isometry group. The generators of SU (2) ∼ SO(3) are the three angular momentum
operators La ,
La = −iεabc xb ∂c ,
which in terms of the usual spherical coordinates θ and φ become
∂
∂
+ i cos φ cot θ ,
∂θ
∂φ
∂
∂
L2 = −i cos φ
+ i sin φ cot θ ,
∂θ
∂φ
∂
L3 = −i .
∂φ
L1 = i sin φ
These relations can be summarized as
La = −iξaα ∂α ,
where the Greek index α corresponds to the spherical coordinates and ξaα are the components of
the Killing vector fields associated with the isometries of the sphere. The metric tensor of the
sphere can be expressed in terms of the Killing vectors as
g αβ =
1 α β
ξ ξ .
R2 a a
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
5
Any function on the sphere can be expanded in terms of the eigenfunctions of the sphere,
a(θ, φ) =
l
∞ X
X
alm Ylm (θ, φ),
(2.1)
l=0 m=−l
where alm is a complex coefficient and Ylm (θ, φ) are the spherical harmonics, which satisfy the
equation
L2 Ylm = −R2 ∆S 2 Ylm = l(l + 1)Ylm ,
where ∆S 2 is the scalar Laplacian on the sphere
1
√
∆S 2 = √ ∂a g ab g∂b .
g
The spherical harmonics have an eigenvalue µ ∼ l(l + 1) for integer l = 0, 1, . . . , with degeneracy
2l + 1. The orthogonality condition of the spherical harmonics is
Z
†
dΩYlm
Yl′ m′ = δll′ δmm′ ,
where dΩ = sin θdθdφ.
The spherical harmonics can be expressed in terms of the cartesian coordinates xa of a unit
vector in R3 ,
X
fa(lm)
xa1 · · · xal
Ylm (θ, φ) =
(2.2)
1 ...al
~a
(lm)
where fa1 ...al is a traceless symmetric tensor of SO(3) with rank l.
Similarly we can expand N × N matrices on a sphere as,
a
ˆ=
N
−1
X
l
X
alm Yˆlm ,
Yˆlm = R−l
l=0 m=−l
X
~a
ˆ a1 · · · X
ˆ al ,
X
fa(lm)
1 ...al
(2.3)
where
ˆ a = √ 2R λa(N )
X
N2 − 1
(2.4)
(N )
(lm)
and λa are the generators of SU (2) in the N -dimensional representation. The tensor faˆ1 ...aˆl is
the same one as in (2.2). The matrices Yˆlm are known as fuzzy spherical harmonics for reasons
which will be apparent shortly. They obey the orthonormality condition
† ˆ
TrN Yˆlm
Yl′ m′ = δll′ δmm′ .
There is an obvious relation between equations (2.1) and (2.3), namely1
a
ˆ=
N
−1
X
l
X
l=0 m=−l
alm Yˆlm
→
a(θ, φ) =
N
−1
X
l
X
alm Ylm (θ, φ).
l=0 m=−l
1
Let us note that in general the map from matrices to functions is not unique, since the expansion coefficients alm may be different. However, here we introduce the fuzzy sphere by truncating the algebra of functions
on the ordinary sphere and therefore the use of the same expansion coefficients is a natural choice.
6
A. Chatzistavrakidis and G. Zoupanos
Notice that the expansion in spherical harmonics is truncated at N − 1 reflecting the finite
number of degrees of freedom in the matrix a
ˆ. This allows the consistent definition of a matrix
approximation of the sphere known as fuzzy sphere.
According to the above discussion the fuzzy sphere [24] is a matrix approximation of the usual
sphere S 2 . The algebra of functions on S 2 (for example spanned by the spherical harmonics)
is truncated at a given frequency and thus becomes finite-dimensional. The truncation has
to be consistent with the associativity of the algebra and this can be nicely achieved relaxing
the commutativity property of the algebra. The fuzzy sphere is the “space” described by this
non-commutative algebra. The algebra itself is that of N × N matrices, which we denote as
2 at fuzziness level N − 1 is the non-commutative
Mat(N ; C). More precisely, the fuzzy sphere SN
ˆ a are N × N hermitian matrices proportional to the
manifold whose coordinate functions X
generatorsP
of the N -dimensional representation of SU (2) as in equation (2.4). They satisfy the
ˆaX
ˆ a = R2 and the commutation relations
condition 3a=1 X
ˆc,
ˆa, X
ˆ b ] = iαεabc X
[X
where α = √N2R2 −1 . It can be proven that for N → ∞ one obtains the usual commutative sphere.
In the following we shall mainly work with the following antihermitian matrices,
Xa =
ˆa
X
,
iαR
which describe equivalently the algebra of the fuzzy sphere and they satisfy the relations
3
X
a=1
Xa Xa = −
1
,
α2
[Xa , Xb ] = Cabc Xc ,
where Cabc = εabc /R.
On the fuzzy sphere there is a natural SU (2) covariant differential calculus. This calculus is
three-dimensional and the derivations ea along Xa of a function f are given by ea (f ) = [Xa , f ].
Accordingly the action of the Lie derivatives on functions is given by
La f = [Xa , f ];
these Lie derivatives satisfy the Leibniz rule and the SU (2) Lie algebra relation
[La , Lb ] = Cabc Lc .
In the N → ∞ limit the derivations ea become ea = Cabc xb ∂ c and only in this commutative
limit the tangent space becomes two-dimensional. The exterior derivative is given by
df = [Xa , f ]θa
with θa the one-forms dual to the vector fields ea , hea , θb i =P
δab . The space of one-forms is
a
generated
θ ’s in the sense that for any one-form ω = i fi dhi ti we can always write
P3 by the
a
ω = a=1 ωa θ with given functions ωa depending on the functions fi , hi and ti . The action of
the Lie derivatives La on the one-forms θb explicitly reads
La (θb ) = Cabc θc .
On a general one-form ω = ωa θa we have Lb ω = Lb (ωa θa ) = [Xb , ωa ] θa − ωa C abc θc and therefore
(Lb ω)a = [Xb , ωa ] − ωc C cba .
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
7
2 , is
The differential geometry on the product space Minkowski times fuzzy sphere, M 4 × SN
4
2
4
2 is
easily obtained from that on M and on SN . For example a one-form A defined on M × SN
written as
A = Aµ dxµ + Aa θa
with Aµ = Aµ (xµ , Xa ) and Aa = Aa (xµ , Xa ).
One can also introduce spinors on the fuzzy sphere and study the Lie derivative on these
spinors [19]. Although here we have sketched the differential geometry on the fuzzy sphere, one
can study other (higher-dimensional) fuzzy spaces (e.g. fuzzy CP M [25], see also [26]) and with
similar techniques their differential geometry.
2.1.1
Gauge theory on the fuzzy sphere
In order to describe gauge fields on the fuzzy sphere it is natural to introduce the notion of
covariant coordinates [27]. In order to do so let us begin with a field φ(Xa ) on the fuzzy sphere,
which is a polynomial in the Xa coordinates. An infinitesimal gauge transformation δφ of the
field φ with gauge transformation parameter λ(Xa ) is defined by
δφ(X) = λ(X)φ(X).
This is an infinitesimal Abelian U (1) gauge transformation if λ(X) is just an antihermitian
function of the coordinates Xa , while it is an infinitesimal non-Abelian U (P ) gauge transformation if λ(X) is valued in Lie(U (P )), the Lie algebra of hermitian P × P matrices. In the
following we will always assume Lie(U (P )) elements to commute with the coordinates Xa . The
coordinates Xa are invariant under a gauge transformation
δXa = 0.
Then, multiplication of a field on the left by a coordinate is not a covariant operation in the
non-commutative case. That is
δ(Xa φ) = Xa λ(X)φ,
and in general the right hand side is not equal to λ(X)Xa φ. Following the ideas of ordinary
gauge theory one then introduces covariant coordinates φa such that
δ(φa φ) = λφa φ.
This happens if
δ(φa ) = [λ, φa ].
(2.5)
The analogy with ordinary gauge theory also suggests to set
φa ≡ Xa + Aa
and interpret Aa as the gauge potential of the non-commutative theory. Then φa is the noncommutative analogue of a covariant derivative. The transformation properties of Aa support
the interpretation of Aa as gauge field, since from requirement (2.5) we can deduce that Aa
transforms as
δAa = −[Xa , λ] + [λ, Aa ].
8
A. Chatzistavrakidis and G. Zoupanos
Correspondingly we can define a tensor Fab , the analogue of the field strength, as
Fab = [Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − C cab Ac = [φa , φb ] − C cab φc .
(2.6)
The presence of the last term in (2.6) might seem strange at first sight, however it is imposed in
the definition of the field strength by the requirement of covariance. Indeed, it is straightforward
to prove that the above tensor transforms covariantly, i.e.
δFab = [λ, Fab ].
Similarly, for a spinor ψ in the adjoint representation, the infinitesimal gauge transformation is
given by
δψ = [λ, ψ].
2.2
Dimensional reduction and gauge symmetry enhancement
Let us now consider a non-commutative gauge theory on M 4 × (S/R)F with gauge group G =
U (P ) and examine its four-dimensional interpretation. (S/R)F is a fuzzy coset, for example the
2 . The action is
fuzzy sphere SN
Z
1
(2.7)
SYM = 2 d4 x kTr trG FM N F M N ,
4g
where kTr denotes integration over the fuzzy coset (S/R)F described by N × N matrices; here
the parameter k is related
to the size of the fuzzy coset space. For example for the fuzzy
√
2
2
sphere we have R = N − 1πk [9]. In the N → ∞ limit kTr becomes the usual integral
on the coset space. For finite N , Tr is a good integral because it has the cyclic property
Tr(f1 · · · fp−1 fp ) = Tr(fp f1 · · · fp−1 ). It is also invariant under the action of the group S, that
is infinitesimally given by the Lie derivative. In the action (2.7) trG is the gauge group G
trace. The higher-dimensional field strength FM N , decomposed in four-dimensional space-time
and extra-dimensional components, reads as (Fµν , Fµb , Fab ), where µ, ν are four-dimensional
spacetime indices. The various components of the field strength are explicitly given by
Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ],
Fµa = ∂µ Aa − [Xa , Aµ ] + [Aµ , Aa ],
Fab = [Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − C cab Ac .
In terms of the covariant coordinates φ, which were introduced in the previous section, the field
strength in the non-commutative directions becomes
Fµa = ∂µ φa + [Aµ , φa ] = Dµ φa ,
Fab = [φa , φb ] − C cab φc .
Using these expressions the action (2.7) becomes
Z
k
k 2
2
4
F +
(Dµ φa ) − V (φ) ,
SYM = d x Tr trG
4g 2 µν 2g 2
(2.8)
where the potential term V (φ) is the Fab kinetic term (in our conventions Fab is antihermitian
so that V (φ) is hermitian and non-negative),
V (φ) = −
X
k
Fab Fab
Tr
tr
G
4g 2
ab
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
=−
k
a b
a b c
−2 2
[φ
,
φ
][φ
,
φ
]
−
4C
φ
φ
φ
+
2R
φ
.
Tr
tr
a
G
b
abc
4g 2
9
(2.9)
The action (2.8) is naturally interpreted as an action in four dimensions. The infinitesimal G
gauge transformation with gauge parameter λ(xµ , X a ) can indeed be interpreted just as an M 4
gauge transformation. We write
λ(xµ , X a ) = λI (xµ , X a )T I = λh,I (xµ )T h T I ,
(2.10)
where T I are hermitian generators of U (P ), λI (xµ , X a ) are N × N antihermitian matrices
and thus are expressible as λ(xµ )I,h T h , where T h are antihermitian generators of U (N ). The
fields λ(xµ )I,h , with h = 1, . . . , N 2 , are the Kaluza–Klein modes of λ(xµ , X a )I . We now consider
on equal footing the indices h and I and interpret the fields on the r.h.s. of (2.10) as one field
valued in the tensor product Lie algebra Lie(U (N )) ⊗ Lie(U (P )). This Lie algebra is indeed
Lie(U (N P )) (the (N P )2 generators T h T I being N P × N P antihermitian matrices that are
linear independent). Similarly we rewrite the gauge field Aν as
µ
h I
Aν (xµ , X a ) = AIν (xµ , X a )T I = Ah,I
ν (x )T T ,
and interpret it as a Lie(U (N P ))-valued gauge field on M 4 . The four-dimensional scalar fields φa
are interpreted similarly. It is worth noting that the scalars transform in the adjoint representation of the four-dimensional gauge group and therefore they are not suitable for the electroweak
symmetry breaking. This serves as a motivation to use a non-trivial dimensional reduction
scheme, which is presented in the following section. Finally Tr trG is the trace over U (N P )
matrices in the fundamental representation.
Up to now we have just performed a ordinary fuzzy dimensional reduction. Indeed in the
commutative case the expression (2.8) corresponds to rewriting the initial lagrangian on M 4 ×S 2
using spherical harmonics on S 2 . Here the space of functions is finite-dimensional and therefore
the infinite tower of modes reduces to the finite sum given by the trace Tr. The remarkable
result of the above analysis is that the gauge group in four dimensions is enhanced compared to
the gauge group G in the higher-dimensional theory. Therefore it is very interesting to note that
we can in fact start with an Abelian gauge group in higher dimensions and obtain non-Abelian
gauge symmetry in the four-dimensional theory.
2.3
Non-trivial dimensional reduction over fuzzy extra dimensions
In this section we reduce the number of gauge fields and scalars in the action (2.8) by applying
the Coset Space Dimensional Reduction (CSDR) scheme. Before proceeding to the case of fuzzy
extra dimensions let us briefly recall how this scheme works in the commutative case.
2.3.1
Ordinary CSDR
One way to dimensionally reduce a gauge theory on M 4 × S/R with gauge group G to a gauge
theory on M 4 , is to consider field configurations that are invariant under S/R transformations.
Since the action of the group S on the coset space S/R is transitive (i.e., connects all points), we
can equivalently require the fields in the theory to be invariant under the action of S on S/R.
Infinitesimally, if we denote by ξa the Killing vectors on S/R associated to the generators T a
of S, we require the fields to have zero Lie derivative along ξa . For scalar fields this is equivalent
to requiring independence under the S/R coordinates. The CSDR scheme dimensionally reduces
a gauge theory on M 4 × S/R with gauge group G to a gauge theory on M 4 imposing a milder
constraint, namely the fields are required to be invariant under the S action up to a G gauge
10
A. Chatzistavrakidis and G. Zoupanos
transformation [3, 4, 5]2 . Thus we have, respectively for scalar fields φ and the one-form gauge
field A,
Lξa φ = δ Wa φ = Wa φ,
Lξa A = δ Wa A = −DWa ,
(2.11)
where δ Wa is the infinitesimal gauge transformation relative to the gauge parameter Wa that
depends on the coset coordinates (in our notations A and Wa are antihermitian and the covariant
derivative reads D = d + A). The gauge parameters Wa obey a consistency condition which
follows from the relation
[Lξa , Lξb ] = L[ξa ,ξb ]
(2.12)
and transform under a gauge transformation φ → gφ as
Wa → gWa g −1 + (Lξa g)g −1 .
(2.13)
Since two points of the coset are connected by an S-transformation which is equivalent to a gauge
transformation, and since the Lagrangian is gauge invariant, we can study the above equations
just at one point of the coset, let’s say y α = 0, where we denote by (xµ , y α ) the coordinates of
M 4 × S/R, and we use a, α, i to denote S, S/R and R indices. In general, using (2.13), not all
the Wa can be gauged transformed to zero at y α = 0, however one can choose Wα = 0 denoting
by Wi the remaining ones. Then the consistency condition which follows from equation (2.12)
implies that Wi are constant and equal to the generators of the embedding of R in G (thus in
particular R must be embeddable in G; we write RG for the image of R in G).
The detailed analysis of the constraints given in [3, 4] provides us with the four-dimensional
unconstrained fields as well as with the gauge invariance that remains in the theory after dimensional reduction. Here we just state the results:
• The components Aµ (x, y) of the initial gauge field AM (x, y) become, after dimensional
reduction, the four-dimensional gauge fields and furthermore they are independent of y.
In addition one can find that they have to commute with the elements of the RG subgroup
of G. Thus the four-dimensional gauge group H is the centralizer of R in G, H = CG (RG ).
• Similarly, the Aα (x, y) components of AM (x, y) denoted by φα (x, y) from now on, become
scalars in four dimensions. These fields transform under R as a vector v, i.e.
S ⊃ R,
adj S = adj R + v.
Moreover φα (x, y) acts as an intertwining operator connecting induced representations of
R acting on G and S/R. This implies, exploiting Schur’s lemma, that the transformation
properties of the fields φα (x, y) under H can be found if we express the adjoint representation of G in terms of RG × H:
G ⊃ RG × H,
X
adj G = (adj R, 1) + (1, adj H) +
(ri , hi ).
P
Then if v = si , where each si is an irreducible representation of R, there survives an hi
multiplet for every pair (ri , si ), where ri and si are identical irreps. of R. If we start from
a pure gauge theory on M 4 × S/R, the four-dimensional potential (at y α = 0) can be
shown to be given by
1
1
V = − Fαβ F αβ = − (C cαβ φc − [φα , φβ ])2 ,
4
4
2
See also [28] for related work.
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
11
where we have defined φi ≡ Wi . However, the fields φα are not independent because
the conditions (2.11) at y α = 0 constrain them. The solution of the constraints provides
the physical dimensionally reduced fields in four dimensions; in terms of these physical
fields the potential is still a quartic polynomial. Then, the minimum of this potential will
determine the spontaneous symmetry breaking pattern.
• Turning next to the fermion fields, similarly to scalars, they act as an intertwining operator
connecting induced representations of R in G and in SO(d), where d is the dimension
of the tangent space of S/R. Proceeding along similar lines as in the case of scalars,
and considering the more interesting case of even dimensions, we impose first the Weyl
condition. Then to obtain the representation of H under which the four-dimensional
fermions transform, we have to decompose the fermion representation F of the initial
gauge group G under RG × H, i.e.
X
F =
(ti , hi ),
and the spinor of SO(d) under R
X
σd =
σj .
Then for each pair ti and σi , where ti and σi are identical irreps. there is an hi multiplet
of spinor fields in the four-dimensional theory. In order however to obtain chiral fermions
in the effective theory we have to impose further requirements [4, 7]. The issue of chiral
fermions will be discussed in Section 2.4.
2.3.2
Fuzzy CSDR
Let us now discuss how the above scheme can be applied in the case where the extra dimensions
are fuzzy coset spaces [19]3 . Since SU (2) acts on the fuzzy sphere (SU (2)/U (1))F , and more
in general the group S acts on the fuzzy coset (S/R)F , we can state the CSDR principle in the
same way as in the continuum case, i.e. the fields in the theory must be invariant under the
infinitesimal SU (2), respectively S, action up to an infinitesimal gauge transformation
Lb φ = δWb φ = Wb φ,
Lb A = δWb A = −DWb ,
(2.14)
(2.15)
where A is the one-form gauge potential A = Aµ dxµ + Aa θa , and Wb depends only on the coset
coordinates X a and (like Aµ , Aa ) is antihermitian. We thus write Wb = WbI T I , I = 1, 2, . . . , P 2 ,
where T I are hermitian generators of U (P ) and (WbI )† = −WbI ; here † is hermitian conjugation
on the X a ’s.
In terms of the covariant coordinate φa = Xa + Aa and of
ωa ≡ Xa − Wa ,
the CSDR constraints (2.14) and (2.15) assume a particularly simple form, namely
[ωb , Aµ ] = 0,
(2.16)
e
(2.17)
Cbde φ = [ωb , φd ].
In addition we have a consistency condition following from the relation [La , Lb ] = Cabc Lc :
[ωa , ωb ] = Cabc ωc ,
(2.18)
where ωa transforms as ωa → ωa′ = gωa g −1 . One proceeds in a similar way for the spinor
fields [19].
3
A similar approach has also been considered in [29].
12
2.3.3
A. Chatzistavrakidis and G. Zoupanos
Solving the CSDR constraints for the fuzzy sphere
2 , i.e. the fuzzy sphere, and to be definite at fuzziness level N −1 (N ×N
We consider (S/R)F = SN
matrices). We study here the basic example where the gauge group is G = U (1). In this case the
ωa = ωa (X b ) appearing in the consistency condition (2.18) are N × N antihermitian matrices
and therefore can be interpreted as elements of Lie(U (N )). On the other hand the ωa satisfy
the commutation relations (2.18) of Lie(SU (2)). Therefore in order to satisfy the consistency
condition (2.18) we have to embed Lie(SU (2)) in Lie(U (N )). Let T h with h = 1, . . . , (N )2 be
the generators of Lie(U (N )) in the fundamental representation. Then we can always use the
convention h = (a, u) with a = 1, 2, 3 and u = 4, 5, . . . , N 2 where the T a satisfy the SU (2) Lie
algebra,
[T a , T b ] = C abc T c .
(2.19)
Then we define an embedding by identifying
ωa = Ta .
(2.20)
The constraint (2.16), [ωb , Aµ ] = 0, then implies that the four-dimensional gauge group K is the
centralizer of the image of SU (2) in U (N ), i.e.
K = CU (N ) (SU ((2))) = SU (N − 2) × U (1) × U (1),
where the last U (1) is the U (1) of U (N ) ≃ SU (N ) × U (1). The functions Aµ (x, X) are arbitrary functions of x but the X dependence is such that Aµ (x, X) is Lie(K)-valued instead
of Lie(U (N )), i.e. eventually we have a four-dimensional gauge potential Aµ (x) with values in
Lie(K). Concerning the constraint (2.17), it is satisfied by choosing
φa = rφ(x)ωa ,
(2.21)
i.e. the unconstrained degrees of freedom correspond to the scalar field φ(x) which is a singlet
under the four-dimensional gauge group K.
The choice (2.20) defines one of the possible embedding of Lie(SU (2)) in Lie(U (N )). For
example, we could also embed Lie(SU (2)) in Lie(U (N )) using the irreducible N -dimensional
rep. of SU (2), i.e. we could identify ωa = Xa . The constraint (2.16) in this case implies that the
four-dimensional gauge group is U (1) so that Aµ (x) is U (1) valued. The constraint (2.17) leads
again to the scalar singlet φ(x).
2 . We solve the CSDR conIn general, we start with a U (1) gauge theory on M 4 × SN
straint (2.18) by embedding SU (2) in U (N ). There exist pN embeddings, where pN is the
number of ways one can partition the integer N into a set of non-increasing positive integers [24]. Then the constraint (2.16) gives the surviving four-dimensional gauge group. The
constraint (2.17) gives the surviving four-dimensional scalars and equation (2.21) is always a solution but in general not the only one. By setting φa = ωa we obtain always a minimum of the
potential. This minimum is given by the chosen embedding of SU (2) in U (N ).
Concerning fermions in the adjoint, the corresponding analysis in [19] shows that we have to
consider the embedding
S ⊂ SO(dim S),
which is given by Ta = 12 Cabc Γbc that satisfies the commutation relation (2.19). Therefore ψ
is an intertwining operator between induced representations of S in U (N P ) and in SO(dim S).
To find the surviving fermions, as in the commutative case [4], we decompose the adjoint rep.
of U (N P ) under SU (N P ) × K,
U (N P ) ⊃ SU (N P ) × K,
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
adj U (N P ) =
X
13
(si , ki ).
i
We also decompose the spinor rep. σ of SO(dim S) under S
SO(dim S) ⊃ S,
X
σ=
σe .
e
Then, when we have two identical irreps. si = σe , there is a ki multiplet of fermions surviving
in four dimensions, i.e. four-dimensional spinors ψ(x) belonging to the ki representation of K.
An important point that we would like to stress here is the question of the renormalizability
of the gauge theory defined on M4 × (S/R)F . First we notice that the theory exhibits certain
features so similar to a higher-dimensional gauge theory defined on M4 × S/R that naturally
it could be considered as a higher-dimensional theory too. For instance the isometries of the
spaces M4 × S/R and M4 × (S/R)F are the same. It does not matter if the compact space
2 , the isometries are
is fuzzy or not. For example in the case of the fuzzy sphere, i.e. M4 × SN
2
SO(3, 1) × SO(3) as in the case of the continuous space, M4 × S . Similarly the coupling of
a gauge theory defined on M4 ×S/R and on M4 ×(S/R)F are both dimensionful and have exactly
the same dimensionality. On the other hand the first theory is clearly non-renormalizable, while
the latter is renormalizable (in the sense that divergencies can be removed by a finite number
of counterterms). So from this point of view one finds a partial justification of the old hopes
for considering quantum field theories on non-commutative structures. If this observation can
lead to finite theories too, it remains as an open question.
2.4
The problem of chirality in fuzzy CSDR
Among the great successes of the ordinary CSDR is the possibility to accommodate chiral
fermions in the four-dimensional theory [7]. Needless to say that the requirement of chirality
for the four-dimensional fermions is necessary in order for a theory to have a chance to become
realistic.
Let us recall the necessary conditions for accommodating chiral fermions in four dimensions
when a higher-dimensional gauge theory with gauge group G is reduced over a d-dimensional
coset space S/R using the CSDR scheme. As we discussed previously, solving the CSDR constraints for the fermion fields leads to the result that in order to obtain the representations
of the four-dimensional unbroken gauge group H under which the four-dimensional fermions
transform, we have to decompose the representation F of the initial gauge group in which the
fermions are assigned under R × H, i.e.
X
F =
(ti , hi ),
and the spinor of the tangent space group SO(d) under R
X
σd =
σj .
Then for each pair ti and σi , where ti and σi are identical irreducible representations there is an
hi multiplet of spinor fields in the four-dimensional theory.
In order to obtain chiral fermions in four dimensions we need some further requirements. The
representation of interest, for our purposes, of the spin group is the spinor representation. This
(d−1)
d
has dimensions 2 2 and 2 2 for d even and odd respectively. For odd d the representation is
irreducible but for even d it is reducible into two irreducible components of equal dimension.
This splitting exactly gives the possibility to define Weyl spinors and to construct a chirality
14
A. Chatzistavrakidis and G. Zoupanos
operator. Thus if we are in odd number of dimensions (where the chirality operator does
not exist) there is no way to obtain chiral fermions. For this reason we focus only on even
dimensions.
The first possibility is to start with Dirac fermions in D (even) dimensions. Here we can
define the standard chirality operator
ΓD+1 = i
D(D−1)
2
Γ1 Γ2 · · · ΓD ,
with (ΓD+1 )2 = 1 and {ΓD+1 , ΓA } = 0, where ΓA , A = 1, . . . , D span the Clifford algebra in D
dimensions. This operator has eigenvalues ±1 and distinguishes left and right spinors. So, it is
possible to define a Weyl basis, where the chirality operator is diagonal, namely
ΓD+1 ψ± = ±ψ± .
As we mentioned above, in this case SO(1, D − 1) has two independent irreducible spinor rep′ , under which the Weyl spinors ψ and ψ transform respectively. The
resentations, σD and σD
+
−
following branching rule for the spinors holds4
SO(1, D − 1) ⊃ SO(1, 3) × SO(d),
σD = (2, 1; σd ) + (1, 2; σd′ ),
′
σD
= (2, 1; σd′ ) + (1, 2; σd ).
Then, since we started with a Dirac spinor ψ = ψ+ ⊕ ψ− transforming under a representation F
of the original gauge group G, following the rule which was stated above it is obvious that we
obtain fermions in four dimensions appearing in equal numbers of left and right representations
of the unbroken gauge group H. Thus, starting with Dirac fermions does not render the fermions
of the four-dimensional theory chiral.
In order to overcome this problem we can make a further restriction and start with Weyl
fermions, namely to impose the Weyl condition in higher dimensions. Then, only one of the σD
′ representations is selected. There are still two cases to investigate, the total number of
and σD
dimensions being 4n or 4n + 2. Since we are interested in vacuum configurations of the form
M 4 × S/R the dimensionality of the internal (coset) space is then of the form 4n or 4n + 2
respectively.
For D = 4n (d = 4(n − 1)), the two spinor representations of SO(d) are self-conjugate,
meaning that in the decomposition
SO(d) ⊃ R,
X
σd =
σi ,
σi is either a real representation or it appears together with its conjugate representation σ
¯i .
Thus we are led to consider that the representation F of G where the fermions are assigned
has to be complex. Two important things to note is that R is also required to admit complex
representations (otherwise the decompositions of σd and σd′ will be the same, leading to a nonchiral theory) and that rank S = rank R (otherwise σd and σd′ will again be the same). These
requirements still hold in the following case.
In the case D = 4n + 2 (d = 4(n − 1) + 2), the two spinor representations of SO(d) are not
self-conjugate anymore and σd′ = σ
¯d . Now, the decomposition reads as
SO(d) ⊃ R,
4
Here the usual notation for two-component Weyl spinors of the Lorentz group SO(1, 3) is adopted, namely
ψ+ → (1, 2) and ψ− → (2, 1).
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
σd =
σ
¯d =
X
X
15
σi ,
σ
¯i ,
so we can let F be a vectorlike representation. Then, in the decomposition
G ⊃ RG × H,
X
F =
(ti , hi ),
¯ i ). According
each term (ti , hi ) will either be self-conjugate or it will appear with the term (t¯i , h
to the established rule, σd will provide P
a left-handed fermion multiplet transforming under the
four-dimensional gaugeP
group as fL =
hL
¯d will provide a right-handed fermion multiplet
i ; σ
¯ R . Since hL ∼ h
¯ R we are led to two Weyl fermions with the same
h
transforming as fR =
i
i
i
chirality in the same representation of the unbroken gauge group H. This is of course a chiral
theory, which is the desired result. Moreover, the doubling of the fermions can be eliminated by
imposing the Majorana condition, if applicable5 .
Let us use the same spirit in order to investigate the possibility of obtaining chiral fermions
in the fuzzy case as well. We discussed previously that we have to consider the embedding
S ⊂ SO(dim S),
concerning fermions in the adjoint. In order to determine the surviving fermions, as in the
commutative case, we decompose the adjoint rep. of U (N ) under SU (N ) × K,
U (N ) ⊃ SU (N ) × K,
X
adj U (N ) =
(si , ki ).
i
We also decompose the spinor rep. σ of SO(dim S) under S
SO(dim S) ⊃ S,
X
σe .
σ=
e
Then, when we have two identical irreps. si = σe , there is a ki multiplet of fermions surviving
in four dimensions, i.e. four-dimensional spinors ψ(x) belonging in the ki representation of K.
Concerning the issue of chirality, the situation is now different. The main difference is
obviously the modification of the rule for the surviving fermions. In the continuous case we had
to embed R in SO(d), while now the suitable embedding is that of S in SO(dim S). Exploring
chirality in the continuous case, we had to deal with the representations of SO(d). Recall that
we required d to be even so that there are two independent spinor representations; therefore
in the fuzzy case we require dim S to be even. Moreover, when d = 4n we concluded that the
representation F , where the fermions are initially assigned, has to be complex. Since in the
fuzzy case we assign the fermions in the adjoint representation, the case dim S = 4n would lead
to a non-chiral theory. Finally, the case dim S = 4n + 2 is the only promising one and one would
expect to obtain chiral fermions, as in the continuous case when d = 4n + 2. However, we also
need the further requirement that S admits complex representations, again in analogy with R
admitting complex representations in the continuous case.
In summary, in order to have a chance to obtain chiral fermions in the case of fuzzy extra
dimensions the necessary requirements are:
5
Let us remind that the Majorana condition can be imposed when the number of dimensions is D = 2, 3, 8n+4.
16
A. Chatzistavrakidis and G. Zoupanos
• dim S = 4n + 2,
• S admits complex irreps.
The above requirements are quite restrictive; for example they are not satisfied in the case of
a single fuzzy sphere. In general, using elementary number theory one can show that they
cannot be satisfied for any S being a SU (n), SO(n) or Sp(n) group. Therefore only products of
fuzzy spaces have a chance to lead to chiral fermions after dimensional reduction without further
requirements. The simplest case which satisfies these requirements is that of a product of two
fuzzy spheres, which will be discussed in Section 3.4 in the context of dynamical generation of
fuzzy extra dimensions.
In conclusion it is worth making the following remark. As we saw above, a major difference
between fuzzy and ordinary CSDR is that in the fuzzy case one always embeds S in the gauge
group G instead of embedding just R in G. A generic feature of the ordinary CSDR in the
special case when S is embedded in G is that the fermions in the final theory are massive [30].
According to the discussion in Section 2.3.3 the situation in the fuzzy case is very similar to the
one we just described. In fuzzy CSDR the spontaneous symmetry breaking mechanism takes
already place by solving the fuzzy CSDR constraints. Therefore in the Yukawa sector of the
theory we have the results of the spontaneous symmetry breaking, i.e. massive fermions and
Yukawa interactions among fermions and the physical Higgs field. We shall revisit the problem
of chirality in the following section and finally, in Section 4, we shall describe a way to overcome
it and obtain chiral four-dimensional theories.
3
Dynamical generation of fuzzy extra dimensions
Let us now discuss a further development [20] of these ideas, which addresses in detail the
questions of quantization and renormalization. This leads to a slightly modified model with
an extra term in the potential, which dynamically selects a unique (nontrivial) vacuum out
of the many possible CSDR solutions, and moreover generates a magnetic flux on the fuzzy
2 .
sphere. It also allows to show that the full tower of Kaluza–Klein modes is generated on SN
Moreover, upon including fermions, the model offers the possibility of a detailed study of the
fermionic sector [21]. Such a study reveals the difficulty in obtaining chiral low-energy models
but at the same time it paves the way out of this problem. Indeed, we shall see in the following
section that using orbifold techniques it is possible to construct chiral models in the framework
of dynamically generated fuzzy extra dimensions.
3.1
The four dimensional action
We start with a SU (N ) gauge theory on four dimensional Minkowski space M 4 with coordinates y µ , µ = 0, 1, 2, 3. The action under consideration is
Z
1
†
†
SY M = d4 y T r
F
+
(D
φ
)
D
φ
F
µν
µ a
µ a − V (φ),
4g 2 µν
where Aµ are SU (N )-valued gauge fields, Dµ = ∂µ + [Aµ , ·], and
φa = −φ†a ,
a = 1, 2, 3
are three antihermitian scalars in the adjoint of SU (N ),
φa → U † φa U,
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
17
where U = U (y) ∈ SU (N ). Furthermore, the φa transform as vectors of an additional global
SO(3) symmetry. The potential V (φ) is taken to be the most general renormalizable action
invariant under the above symmetries, which is
V (φ) = Tr (g1 φa φa φb φb + g2 φa φb φa φb − g3 εabc φa φb φc + g4 φa φa )
g5
g6
+
Tr (φa φa ) Tr (φb φb ) +
Tr(φa φb ) Tr (φa φb ) + g7 .
(3.1)
N
N
This may not look very transparent at first sight, however it can be written in a very intuitive
way. First, we make the scalars dimensionless by rescaling
φ′a = Rφa ,
where R has dimension of length; we will usually suppress R since it can immediately be reinserted, and drop the prime from now on. Now observe that for a suitable choice of R,
R=
2g2
,
g3
the potential can be rewritten as
1 †
h
2
2
˜
V (φ) = Tr a (φa φa + b1l) + c + 2 Fab Fab + gab gab
g˜
N
for suitable constants a, b, c, g˜, h, where
˜b = b + d Tr (φa φa ),
gab = Tr(φa φb ).
N
We will omit c from now. Notice that two couplings were reabsorbed in the definitions of R
and ˜b. The potential is clearly positive definite provided
Fab = [φa , φb ] − εabc φc = εabc Fc ,
a2 = g1 + g2 > 0,
2
= −g2 > 0,
g˜2
h ≥ 0,
which we assume from now on. Here ˜b = ˜b(y) is a scalar, gab = gab (y) is a symmetric tensor
under the
SIGMA 6 (2010), 063, 47 pages
Higher-Dimensional Unif ied Theories
with Fuzzy Extra Dimensions⋆
Athanasios CHATZISTAVRAKIDIS
†‡
and George ZOUPANOS
‡
†
Institute of Nuclear Physics, NCSR Demokritos, GR-15310 Athens, Greece
E-mail: [email protected]
‡
Physics Department, National Technical University of Athens,
GR-15780 Zografou Campus, Athens, Greece
E-mail: [email protected]
Received May 06, 2010, in final form July 22, 2010; Published online August 12, 2010
doi:10.3842/SIGMA.2010.063
Abstract. Theories defined in higher than four dimensions have been used in various
frameworks and have a long and interesting history. Here we review certain attempts,
developed over the last years, towards the construction of unified particle physics models in
the context of higher-dimensional gauge theories with non-commutative extra dimensions.
These ideas have been developed in two complementary ways, namely (i) starting with
a higher-dimensional gauge theory and dimensionally reducing it to four dimensions over
fuzzy internal spaces and (ii) starting with a four-dimensional, renormalizable gauge theory
and dynamically generating fuzzy extra dimensions. We describe the above approaches and
moreover we discuss the inclusion of fermions and the construction of realistic chiral theories
in this context.
Key words: fuzzy extra dimensions; unified gauge theories; symmetry breaking
2010 Mathematics Subject Classification: 70S15
Contents
1 Introduction
2
2 Fuzzy spaces and dimensional reduction
2.1 The fuzzy sphere . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Gauge theory on the fuzzy sphere . . . . . . . . . . . .
2.2 Dimensional reduction and gauge symmetry enhancement . .
2.3 Non-trivial dimensional reduction over fuzzy extra dimensions
2.3.1 Ordinary CSDR . . . . . . . . . . . . . . . . . . . . .
2.3.2 Fuzzy CSDR . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Solving the CSDR constraints for the fuzzy sphere . .
2.4 The problem of chirality in fuzzy CSDR . . . . . . . . . . . .
3 Dynamical generation of fuzzy extra dimensions
3.1 The four dimensional action . . . . . . . . . . . . . .
3.2 Emergence of extra dimensions and the fuzzy sphere
3.3 Inclusion of fermions . . . . . . . . . . . . . . . . . .
2 . . . . .
3.3.1 Fermions on M 4 × S 2 and M 4 × SN
3.3.2 The spectrum of D
6 (2) in a type I vacuum . .
⋆
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16
16
17
19
19
20
This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is
available at http://www.emis.de/journals/SIGMA/noncommutative.html
2
A. Chatzistavrakidis and G. Zoupanos
3.4
3.3.3 The spectrum of D
6 (2) in a type II vacuum . . . . . .
Dynamical generation of fuzzy S 2 × S 2 and mirror fermions
3.4.1 The action . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Type I vacuum and fuzzy S 2 × S 2 . . . . . . . . . .
2 . . . . . . . . . . . . . . . . .
3.4.3 Operators on SL2 × SR
3.4.4 Type II vacuum and the zero-modes . . . . . . . . .
4 Orbifolds, fuzzy extra dimensions and chiral models
4.1 N = 4 SYM and Z3 orbifolds . . . . . . . . . . . . . .
4.2 Twisted fuzzy spheres . . . . . . . . . . . . . . . . . .
4.3 Dynamical generation of twisted fuzzy spheres . . . .
4.4 Chiral models from the fuzzy orbifold . . . . . . . . .
4.4.1 A SU (4)c × SU (2)L × SU (2)R model . . . . . .
4.4.2 A SU (4)c × SU (4)L × SU (4)R model . . . . . .
4.4.3 A SU (3)c × SU (3)L × SU (3)R model . . . . . .
4.4.4 A closer look at the masses . . . . . . . . . . .
4.5 Fuzzy breaking for SU (3)3 . . . . . . . . . . . . . . . .
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21
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21
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24
26
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28
28
30
32
33
34
35
36
37
37
5 Discussion and conclusions
40
A Clif ford algebra conventions
42
References
42
1
Introduction
The unification of the fundamental interactions has always been one of the main goals of theoretical physics. Several approaches have been employed in order to achieve this goal, one of the
most exciting ones being the proposal that extra dimensions may exist in nature. The most
serious support on the existence of extra dimensions came from superstring theories [1], which
at present are the best candidates for a unified description of all fundamental interactions, including gravity and moreover they can be consistently defined only in higher dimensions. Among
superstring theories the heterotic string [2] has always been considered as the most promising
version in the prospect to find contact with low-energy physics studied in accelerators, mainly
due to the presence of the ten-dimensional N = 1 gauge sector. Upon compactification of
the ten-dimensional space-time and subsequent dimensional reduction the initial E8 × E8 gauge
theory can break to phenomenologically interesting Grand Unified Theories (GUTs), where
the Standard Model (SM) could in principle be accommodated [2]. Dimensional reduction of
higher-dimensional gauge theories had been studied few years earlier than the discovery of the
heterotic superstring with pioneer studies the Forgacs–Manton Coset Space Dimensional Reduction (CSDR) [3, 4, 5] and the Scherk–Schwarz group manifold reduction [6]. In these frameworks
gauge-Higgs unification is achieved in higher dimensions, since the four-dimensional gauge and
Higgs fields are simply the surviving components of the gauge fields of a pure gauge theory
defined in higher dimensions. Moreover in the CSDR the addition of fermions in the higherdimensional gauge theory leads naturally to Yukawa couplings in four dimensions. A major
achievement in this direction is the possibility to obtain chiral theories in four dimensions [7].
On the other hand, non-commutative geometry offers another framework aiming to describe
physics at the Planck scale [8, 9]. In the spirit of non-commutative geometry also particle
models with non-commutative gauge theory were explored [10] (see also [11]), [12, 13]. It is
worth stressing the observation that a natural realization of non-commutativity of space appears
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
3
in the string theory context of D-branes in the presence of a constant antisymmetric field [14],
which not only brought together the two approaches but they can be considered complementary.
Another interesting development in the non-commutative framework was the work of Seiberg and
Witten [15], where a map between the non-commutative and commutative gauge theories has
been described. Based on that and related subsequent developments [16, 17] a non-commutative
version of the SM has been constructed [18]. These non-commutative models represent interesting generalizations of the SM and hint at possible new physics. However they do not address
the usual problem of the SM, the presence of a plethora of free parameters mostly related to the
ad hoc introduction of the Higgs and Yukawa sectors in the theory.
According to the above discussion it is natural to investigate higher-dimensional gauge theories and their dimensional reduction in four dimensions. Our aim is to provide an up to-date
overview of certain attempts in this direction, developed over the last years. The development
of these ideas has followed two complementary ways, namely (i) the dimensional reduction
of a higher-dimensional gauge theory over fuzzy internal spaces [19] and (ii) the dynamical
generation of fuzzy extra dimensions within a four-dimensional and renormalizable gauge theory [20, 21, 22, 23].
More specifically, the paper is organized as follows. In Section 2 we present a study of
the CSDR in the non-commutative context which sets the rules for constructing new particle
models that might be phenomenologically interesting. One could study CSDR with the whole
parent space M D being non-commutative or with just non-commutative Minkowski space or noncommutative internal space. We specialize here to this last situation and therefore eventually
we obtain Lorentz covariant theories on commutative Minkowski space. We further specialize
to fuzzy non-commutativity, i.e. to matrix type non-commutativity. Thus, following [19], we
consider non-commutative spaces like those studied in [9, 12, 13] and implementing the CSDR
principle on these spaces we obtain the rules for constructing new particle models. In Section 2.1
the fuzzy sphere is introduced and moreover the gauge theory on the fuzzy sphere is discussed.
In Section 2.2 a trivial dimensional reduction of a higher-dimensional gauge theory over the
fuzzy sphere is performed. In Section 2.3 we discuss the non-trivial dimensional reduction; first
the CSDR scheme in the commutative case is briefly reviewed and subsequently it is applied to
the case of fuzzy extra dimensions. In Section 2.4 the issue of chirality is discussed within the
above context.
In Section 3 we reverse the above approach [20] and examine how a four-dimensional gauge
theory dynamically develops fuzzy extra dimensions. In Sections 3.1 and 3.2 we present a simple
field-theoretical model which realizes the above ideas. It is defined as a renormalizable SU (N )
gauge theory on four-dimensional Minkowski space M 4 , containing three scalars in the adjoint
of SU (N ) that transform as vectors under an additional global SO(3) symmetry with the most
general renormalizable potential. We then show that the model dynamically develops fuzzy
2 . The appropriate interpretation is therefore
extra dimensions, more precisely a fuzzy sphere SN
2 . The low-energy effective action is that of a four-dimensional
as gauge theory on M 4 × SN
4
gauge theory on M , whose gauge group and field content is dynamically determined by com2 . An interesting and rich
pactification and dimensional reduction on the internal sphere SN
pattern of spontaneous symmetry breaking appears, namely the breaking of the original SU (N )
gauge symmetry down to either SU (n) or SU (n1 ) × SU (n2 ) × U (1). The latter case is the
generic one, and implies also a monopole flux induced on the fuzzy sphere. Moreover we
determine explicitly the tower of massive Kaluza–Klein modes corresponding to the effective
geometry, which justifies the interpretation as a compactified higher-dimensional gauge theory.
Last but not least, the model is renormalizable. In Sections 3.3 and 3.4 we explore the dynamical generation of a product of two fuzzy spheres [22]. Specifically, we start with the SU (N )
Yang–Mills theory in four dimensions, coupled to six scalars and four Majorana spinors, i.e.
with the particle spectrum of the N = 4 supersymmetric Yang–Mills theory (SYM). Adding
4
A. Chatzistavrakidis and G. Zoupanos
an explicit R-symmetry-breaking potential, thus breaking the N = 4 supersymmetry, we reveal
2 vacua. In the most interesting case we include magnetic fluxes on the
stable M 4 × SL2 × SR
extra-dimensional fuzzy spheres and study the fermion spectrum, in particular the zero modes
of the Dirac operator. The outcome of our analysis is that we obtain a mirror model in low
energies.
In Section 4 we present a recently developed approach within the above framework, which
leads to chiral low-energy models [23]. In particular, Z3 orbifolds of N = 4 supersymmetric
Yang–Mills theory are discussed and they are subsequently used to dynamically generate fuzzy
extra dimensions. The extra dimensions are described by twisted fuzzy spheres, defined in
Section 4.2. This framework allows to construct low-energy models with interesting unification
groups and a chiral spectrum. In particular, we are led to study three different models based on
the gauge groups SU (4) × SU (2) × SU (2), SU (4)3 and SU (3)3 respectively. The spontaneous
symmetry breaking of the latter unified gauge group down to the minimal supersymmetric
standard model and to the SU (3) × U (1)em is subsequently studied within the same framework.
Finally, Section 5 contains our conclusions.
2
2.1
Fuzzy spaces and dimensional reduction
The fuzzy sphere
The fuzzy sphere [24] is a noncommutative manifold which corresponds to a matrix approximation of the ordinary sphere. In order to describe it let us consider the ordinary sphere as
a submanifold of the three-dimensional Euclidean space R3 . The coordinates of R3 will be denoted as xa , a = 1, 2, 3. Then the algebra of functions on the ordinary sphere S 2 ⊂ R3 can be
generated by the coordinates of R3 modulo the relation
3
X
xa xa = R2 ,
a=1
where R is the radius of the sphere. Clearly, the sphere admits the action of a global SO(3) ∼
SU (2) isometry group. The generators of SU (2) ∼ SO(3) are the three angular momentum
operators La ,
La = −iεabc xb ∂c ,
which in terms of the usual spherical coordinates θ and φ become
∂
∂
+ i cos φ cot θ ,
∂θ
∂φ
∂
∂
L2 = −i cos φ
+ i sin φ cot θ ,
∂θ
∂φ
∂
L3 = −i .
∂φ
L1 = i sin φ
These relations can be summarized as
La = −iξaα ∂α ,
where the Greek index α corresponds to the spherical coordinates and ξaα are the components of
the Killing vector fields associated with the isometries of the sphere. The metric tensor of the
sphere can be expressed in terms of the Killing vectors as
g αβ =
1 α β
ξ ξ .
R2 a a
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
5
Any function on the sphere can be expanded in terms of the eigenfunctions of the sphere,
a(θ, φ) =
l
∞ X
X
alm Ylm (θ, φ),
(2.1)
l=0 m=−l
where alm is a complex coefficient and Ylm (θ, φ) are the spherical harmonics, which satisfy the
equation
L2 Ylm = −R2 ∆S 2 Ylm = l(l + 1)Ylm ,
where ∆S 2 is the scalar Laplacian on the sphere
1
√
∆S 2 = √ ∂a g ab g∂b .
g
The spherical harmonics have an eigenvalue µ ∼ l(l + 1) for integer l = 0, 1, . . . , with degeneracy
2l + 1. The orthogonality condition of the spherical harmonics is
Z
†
dΩYlm
Yl′ m′ = δll′ δmm′ ,
where dΩ = sin θdθdφ.
The spherical harmonics can be expressed in terms of the cartesian coordinates xa of a unit
vector in R3 ,
X
fa(lm)
xa1 · · · xal
Ylm (θ, φ) =
(2.2)
1 ...al
~a
(lm)
where fa1 ...al is a traceless symmetric tensor of SO(3) with rank l.
Similarly we can expand N × N matrices on a sphere as,
a
ˆ=
N
−1
X
l
X
alm Yˆlm ,
Yˆlm = R−l
l=0 m=−l
X
~a
ˆ a1 · · · X
ˆ al ,
X
fa(lm)
1 ...al
(2.3)
where
ˆ a = √ 2R λa(N )
X
N2 − 1
(2.4)
(N )
(lm)
and λa are the generators of SU (2) in the N -dimensional representation. The tensor faˆ1 ...aˆl is
the same one as in (2.2). The matrices Yˆlm are known as fuzzy spherical harmonics for reasons
which will be apparent shortly. They obey the orthonormality condition
† ˆ
TrN Yˆlm
Yl′ m′ = δll′ δmm′ .
There is an obvious relation between equations (2.1) and (2.3), namely1
a
ˆ=
N
−1
X
l
X
l=0 m=−l
alm Yˆlm
→
a(θ, φ) =
N
−1
X
l
X
alm Ylm (θ, φ).
l=0 m=−l
1
Let us note that in general the map from matrices to functions is not unique, since the expansion coefficients alm may be different. However, here we introduce the fuzzy sphere by truncating the algebra of functions
on the ordinary sphere and therefore the use of the same expansion coefficients is a natural choice.
6
A. Chatzistavrakidis and G. Zoupanos
Notice that the expansion in spherical harmonics is truncated at N − 1 reflecting the finite
number of degrees of freedom in the matrix a
ˆ. This allows the consistent definition of a matrix
approximation of the sphere known as fuzzy sphere.
According to the above discussion the fuzzy sphere [24] is a matrix approximation of the usual
sphere S 2 . The algebra of functions on S 2 (for example spanned by the spherical harmonics)
is truncated at a given frequency and thus becomes finite-dimensional. The truncation has
to be consistent with the associativity of the algebra and this can be nicely achieved relaxing
the commutativity property of the algebra. The fuzzy sphere is the “space” described by this
non-commutative algebra. The algebra itself is that of N × N matrices, which we denote as
2 at fuzziness level N − 1 is the non-commutative
Mat(N ; C). More precisely, the fuzzy sphere SN
ˆ a are N × N hermitian matrices proportional to the
manifold whose coordinate functions X
generatorsP
of the N -dimensional representation of SU (2) as in equation (2.4). They satisfy the
ˆaX
ˆ a = R2 and the commutation relations
condition 3a=1 X
ˆc,
ˆa, X
ˆ b ] = iαεabc X
[X
where α = √N2R2 −1 . It can be proven that for N → ∞ one obtains the usual commutative sphere.
In the following we shall mainly work with the following antihermitian matrices,
Xa =
ˆa
X
,
iαR
which describe equivalently the algebra of the fuzzy sphere and they satisfy the relations
3
X
a=1
Xa Xa = −
1
,
α2
[Xa , Xb ] = Cabc Xc ,
where Cabc = εabc /R.
On the fuzzy sphere there is a natural SU (2) covariant differential calculus. This calculus is
three-dimensional and the derivations ea along Xa of a function f are given by ea (f ) = [Xa , f ].
Accordingly the action of the Lie derivatives on functions is given by
La f = [Xa , f ];
these Lie derivatives satisfy the Leibniz rule and the SU (2) Lie algebra relation
[La , Lb ] = Cabc Lc .
In the N → ∞ limit the derivations ea become ea = Cabc xb ∂ c and only in this commutative
limit the tangent space becomes two-dimensional. The exterior derivative is given by
df = [Xa , f ]θa
with θa the one-forms dual to the vector fields ea , hea , θb i =P
δab . The space of one-forms is
a
generated
θ ’s in the sense that for any one-form ω = i fi dhi ti we can always write
P3 by the
a
ω = a=1 ωa θ with given functions ωa depending on the functions fi , hi and ti . The action of
the Lie derivatives La on the one-forms θb explicitly reads
La (θb ) = Cabc θc .
On a general one-form ω = ωa θa we have Lb ω = Lb (ωa θa ) = [Xb , ωa ] θa − ωa C abc θc and therefore
(Lb ω)a = [Xb , ωa ] − ωc C cba .
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
7
2 , is
The differential geometry on the product space Minkowski times fuzzy sphere, M 4 × SN
4
2
4
2 is
easily obtained from that on M and on SN . For example a one-form A defined on M × SN
written as
A = Aµ dxµ + Aa θa
with Aµ = Aµ (xµ , Xa ) and Aa = Aa (xµ , Xa ).
One can also introduce spinors on the fuzzy sphere and study the Lie derivative on these
spinors [19]. Although here we have sketched the differential geometry on the fuzzy sphere, one
can study other (higher-dimensional) fuzzy spaces (e.g. fuzzy CP M [25], see also [26]) and with
similar techniques their differential geometry.
2.1.1
Gauge theory on the fuzzy sphere
In order to describe gauge fields on the fuzzy sphere it is natural to introduce the notion of
covariant coordinates [27]. In order to do so let us begin with a field φ(Xa ) on the fuzzy sphere,
which is a polynomial in the Xa coordinates. An infinitesimal gauge transformation δφ of the
field φ with gauge transformation parameter λ(Xa ) is defined by
δφ(X) = λ(X)φ(X).
This is an infinitesimal Abelian U (1) gauge transformation if λ(X) is just an antihermitian
function of the coordinates Xa , while it is an infinitesimal non-Abelian U (P ) gauge transformation if λ(X) is valued in Lie(U (P )), the Lie algebra of hermitian P × P matrices. In the
following we will always assume Lie(U (P )) elements to commute with the coordinates Xa . The
coordinates Xa are invariant under a gauge transformation
δXa = 0.
Then, multiplication of a field on the left by a coordinate is not a covariant operation in the
non-commutative case. That is
δ(Xa φ) = Xa λ(X)φ,
and in general the right hand side is not equal to λ(X)Xa φ. Following the ideas of ordinary
gauge theory one then introduces covariant coordinates φa such that
δ(φa φ) = λφa φ.
This happens if
δ(φa ) = [λ, φa ].
(2.5)
The analogy with ordinary gauge theory also suggests to set
φa ≡ Xa + Aa
and interpret Aa as the gauge potential of the non-commutative theory. Then φa is the noncommutative analogue of a covariant derivative. The transformation properties of Aa support
the interpretation of Aa as gauge field, since from requirement (2.5) we can deduce that Aa
transforms as
δAa = −[Xa , λ] + [λ, Aa ].
8
A. Chatzistavrakidis and G. Zoupanos
Correspondingly we can define a tensor Fab , the analogue of the field strength, as
Fab = [Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − C cab Ac = [φa , φb ] − C cab φc .
(2.6)
The presence of the last term in (2.6) might seem strange at first sight, however it is imposed in
the definition of the field strength by the requirement of covariance. Indeed, it is straightforward
to prove that the above tensor transforms covariantly, i.e.
δFab = [λ, Fab ].
Similarly, for a spinor ψ in the adjoint representation, the infinitesimal gauge transformation is
given by
δψ = [λ, ψ].
2.2
Dimensional reduction and gauge symmetry enhancement
Let us now consider a non-commutative gauge theory on M 4 × (S/R)F with gauge group G =
U (P ) and examine its four-dimensional interpretation. (S/R)F is a fuzzy coset, for example the
2 . The action is
fuzzy sphere SN
Z
1
(2.7)
SYM = 2 d4 x kTr trG FM N F M N ,
4g
where kTr denotes integration over the fuzzy coset (S/R)F described by N × N matrices; here
the parameter k is related
to the size of the fuzzy coset space. For example for the fuzzy
√
2
2
sphere we have R = N − 1πk [9]. In the N → ∞ limit kTr becomes the usual integral
on the coset space. For finite N , Tr is a good integral because it has the cyclic property
Tr(f1 · · · fp−1 fp ) = Tr(fp f1 · · · fp−1 ). It is also invariant under the action of the group S, that
is infinitesimally given by the Lie derivative. In the action (2.7) trG is the gauge group G
trace. The higher-dimensional field strength FM N , decomposed in four-dimensional space-time
and extra-dimensional components, reads as (Fµν , Fµb , Fab ), where µ, ν are four-dimensional
spacetime indices. The various components of the field strength are explicitly given by
Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ],
Fµa = ∂µ Aa − [Xa , Aµ ] + [Aµ , Aa ],
Fab = [Xa , Ab ] − [Xb , Aa ] + [Aa , Ab ] − C cab Ac .
In terms of the covariant coordinates φ, which were introduced in the previous section, the field
strength in the non-commutative directions becomes
Fµa = ∂µ φa + [Aµ , φa ] = Dµ φa ,
Fab = [φa , φb ] − C cab φc .
Using these expressions the action (2.7) becomes
Z
k
k 2
2
4
F +
(Dµ φa ) − V (φ) ,
SYM = d x Tr trG
4g 2 µν 2g 2
(2.8)
where the potential term V (φ) is the Fab kinetic term (in our conventions Fab is antihermitian
so that V (φ) is hermitian and non-negative),
V (φ) = −
X
k
Fab Fab
Tr
tr
G
4g 2
ab
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
=−
k
a b
a b c
−2 2
[φ
,
φ
][φ
,
φ
]
−
4C
φ
φ
φ
+
2R
φ
.
Tr
tr
a
G
b
abc
4g 2
9
(2.9)
The action (2.8) is naturally interpreted as an action in four dimensions. The infinitesimal G
gauge transformation with gauge parameter λ(xµ , X a ) can indeed be interpreted just as an M 4
gauge transformation. We write
λ(xµ , X a ) = λI (xµ , X a )T I = λh,I (xµ )T h T I ,
(2.10)
where T I are hermitian generators of U (P ), λI (xµ , X a ) are N × N antihermitian matrices
and thus are expressible as λ(xµ )I,h T h , where T h are antihermitian generators of U (N ). The
fields λ(xµ )I,h , with h = 1, . . . , N 2 , are the Kaluza–Klein modes of λ(xµ , X a )I . We now consider
on equal footing the indices h and I and interpret the fields on the r.h.s. of (2.10) as one field
valued in the tensor product Lie algebra Lie(U (N )) ⊗ Lie(U (P )). This Lie algebra is indeed
Lie(U (N P )) (the (N P )2 generators T h T I being N P × N P antihermitian matrices that are
linear independent). Similarly we rewrite the gauge field Aν as
µ
h I
Aν (xµ , X a ) = AIν (xµ , X a )T I = Ah,I
ν (x )T T ,
and interpret it as a Lie(U (N P ))-valued gauge field on M 4 . The four-dimensional scalar fields φa
are interpreted similarly. It is worth noting that the scalars transform in the adjoint representation of the four-dimensional gauge group and therefore they are not suitable for the electroweak
symmetry breaking. This serves as a motivation to use a non-trivial dimensional reduction
scheme, which is presented in the following section. Finally Tr trG is the trace over U (N P )
matrices in the fundamental representation.
Up to now we have just performed a ordinary fuzzy dimensional reduction. Indeed in the
commutative case the expression (2.8) corresponds to rewriting the initial lagrangian on M 4 ×S 2
using spherical harmonics on S 2 . Here the space of functions is finite-dimensional and therefore
the infinite tower of modes reduces to the finite sum given by the trace Tr. The remarkable
result of the above analysis is that the gauge group in four dimensions is enhanced compared to
the gauge group G in the higher-dimensional theory. Therefore it is very interesting to note that
we can in fact start with an Abelian gauge group in higher dimensions and obtain non-Abelian
gauge symmetry in the four-dimensional theory.
2.3
Non-trivial dimensional reduction over fuzzy extra dimensions
In this section we reduce the number of gauge fields and scalars in the action (2.8) by applying
the Coset Space Dimensional Reduction (CSDR) scheme. Before proceeding to the case of fuzzy
extra dimensions let us briefly recall how this scheme works in the commutative case.
2.3.1
Ordinary CSDR
One way to dimensionally reduce a gauge theory on M 4 × S/R with gauge group G to a gauge
theory on M 4 , is to consider field configurations that are invariant under S/R transformations.
Since the action of the group S on the coset space S/R is transitive (i.e., connects all points), we
can equivalently require the fields in the theory to be invariant under the action of S on S/R.
Infinitesimally, if we denote by ξa the Killing vectors on S/R associated to the generators T a
of S, we require the fields to have zero Lie derivative along ξa . For scalar fields this is equivalent
to requiring independence under the S/R coordinates. The CSDR scheme dimensionally reduces
a gauge theory on M 4 × S/R with gauge group G to a gauge theory on M 4 imposing a milder
constraint, namely the fields are required to be invariant under the S action up to a G gauge
10
A. Chatzistavrakidis and G. Zoupanos
transformation [3, 4, 5]2 . Thus we have, respectively for scalar fields φ and the one-form gauge
field A,
Lξa φ = δ Wa φ = Wa φ,
Lξa A = δ Wa A = −DWa ,
(2.11)
where δ Wa is the infinitesimal gauge transformation relative to the gauge parameter Wa that
depends on the coset coordinates (in our notations A and Wa are antihermitian and the covariant
derivative reads D = d + A). The gauge parameters Wa obey a consistency condition which
follows from the relation
[Lξa , Lξb ] = L[ξa ,ξb ]
(2.12)
and transform under a gauge transformation φ → gφ as
Wa → gWa g −1 + (Lξa g)g −1 .
(2.13)
Since two points of the coset are connected by an S-transformation which is equivalent to a gauge
transformation, and since the Lagrangian is gauge invariant, we can study the above equations
just at one point of the coset, let’s say y α = 0, where we denote by (xµ , y α ) the coordinates of
M 4 × S/R, and we use a, α, i to denote S, S/R and R indices. In general, using (2.13), not all
the Wa can be gauged transformed to zero at y α = 0, however one can choose Wα = 0 denoting
by Wi the remaining ones. Then the consistency condition which follows from equation (2.12)
implies that Wi are constant and equal to the generators of the embedding of R in G (thus in
particular R must be embeddable in G; we write RG for the image of R in G).
The detailed analysis of the constraints given in [3, 4] provides us with the four-dimensional
unconstrained fields as well as with the gauge invariance that remains in the theory after dimensional reduction. Here we just state the results:
• The components Aµ (x, y) of the initial gauge field AM (x, y) become, after dimensional
reduction, the four-dimensional gauge fields and furthermore they are independent of y.
In addition one can find that they have to commute with the elements of the RG subgroup
of G. Thus the four-dimensional gauge group H is the centralizer of R in G, H = CG (RG ).
• Similarly, the Aα (x, y) components of AM (x, y) denoted by φα (x, y) from now on, become
scalars in four dimensions. These fields transform under R as a vector v, i.e.
S ⊃ R,
adj S = adj R + v.
Moreover φα (x, y) acts as an intertwining operator connecting induced representations of
R acting on G and S/R. This implies, exploiting Schur’s lemma, that the transformation
properties of the fields φα (x, y) under H can be found if we express the adjoint representation of G in terms of RG × H:
G ⊃ RG × H,
X
adj G = (adj R, 1) + (1, adj H) +
(ri , hi ).
P
Then if v = si , where each si is an irreducible representation of R, there survives an hi
multiplet for every pair (ri , si ), where ri and si are identical irreps. of R. If we start from
a pure gauge theory on M 4 × S/R, the four-dimensional potential (at y α = 0) can be
shown to be given by
1
1
V = − Fαβ F αβ = − (C cαβ φc − [φα , φβ ])2 ,
4
4
2
See also [28] for related work.
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
11
where we have defined φi ≡ Wi . However, the fields φα are not independent because
the conditions (2.11) at y α = 0 constrain them. The solution of the constraints provides
the physical dimensionally reduced fields in four dimensions; in terms of these physical
fields the potential is still a quartic polynomial. Then, the minimum of this potential will
determine the spontaneous symmetry breaking pattern.
• Turning next to the fermion fields, similarly to scalars, they act as an intertwining operator
connecting induced representations of R in G and in SO(d), where d is the dimension
of the tangent space of S/R. Proceeding along similar lines as in the case of scalars,
and considering the more interesting case of even dimensions, we impose first the Weyl
condition. Then to obtain the representation of H under which the four-dimensional
fermions transform, we have to decompose the fermion representation F of the initial
gauge group G under RG × H, i.e.
X
F =
(ti , hi ),
and the spinor of SO(d) under R
X
σd =
σj .
Then for each pair ti and σi , where ti and σi are identical irreps. there is an hi multiplet
of spinor fields in the four-dimensional theory. In order however to obtain chiral fermions
in the effective theory we have to impose further requirements [4, 7]. The issue of chiral
fermions will be discussed in Section 2.4.
2.3.2
Fuzzy CSDR
Let us now discuss how the above scheme can be applied in the case where the extra dimensions
are fuzzy coset spaces [19]3 . Since SU (2) acts on the fuzzy sphere (SU (2)/U (1))F , and more
in general the group S acts on the fuzzy coset (S/R)F , we can state the CSDR principle in the
same way as in the continuum case, i.e. the fields in the theory must be invariant under the
infinitesimal SU (2), respectively S, action up to an infinitesimal gauge transformation
Lb φ = δWb φ = Wb φ,
Lb A = δWb A = −DWb ,
(2.14)
(2.15)
where A is the one-form gauge potential A = Aµ dxµ + Aa θa , and Wb depends only on the coset
coordinates X a and (like Aµ , Aa ) is antihermitian. We thus write Wb = WbI T I , I = 1, 2, . . . , P 2 ,
where T I are hermitian generators of U (P ) and (WbI )† = −WbI ; here † is hermitian conjugation
on the X a ’s.
In terms of the covariant coordinate φa = Xa + Aa and of
ωa ≡ Xa − Wa ,
the CSDR constraints (2.14) and (2.15) assume a particularly simple form, namely
[ωb , Aµ ] = 0,
(2.16)
e
(2.17)
Cbde φ = [ωb , φd ].
In addition we have a consistency condition following from the relation [La , Lb ] = Cabc Lc :
[ωa , ωb ] = Cabc ωc ,
(2.18)
where ωa transforms as ωa → ωa′ = gωa g −1 . One proceeds in a similar way for the spinor
fields [19].
3
A similar approach has also been considered in [29].
12
2.3.3
A. Chatzistavrakidis and G. Zoupanos
Solving the CSDR constraints for the fuzzy sphere
2 , i.e. the fuzzy sphere, and to be definite at fuzziness level N −1 (N ×N
We consider (S/R)F = SN
matrices). We study here the basic example where the gauge group is G = U (1). In this case the
ωa = ωa (X b ) appearing in the consistency condition (2.18) are N × N antihermitian matrices
and therefore can be interpreted as elements of Lie(U (N )). On the other hand the ωa satisfy
the commutation relations (2.18) of Lie(SU (2)). Therefore in order to satisfy the consistency
condition (2.18) we have to embed Lie(SU (2)) in Lie(U (N )). Let T h with h = 1, . . . , (N )2 be
the generators of Lie(U (N )) in the fundamental representation. Then we can always use the
convention h = (a, u) with a = 1, 2, 3 and u = 4, 5, . . . , N 2 where the T a satisfy the SU (2) Lie
algebra,
[T a , T b ] = C abc T c .
(2.19)
Then we define an embedding by identifying
ωa = Ta .
(2.20)
The constraint (2.16), [ωb , Aµ ] = 0, then implies that the four-dimensional gauge group K is the
centralizer of the image of SU (2) in U (N ), i.e.
K = CU (N ) (SU ((2))) = SU (N − 2) × U (1) × U (1),
where the last U (1) is the U (1) of U (N ) ≃ SU (N ) × U (1). The functions Aµ (x, X) are arbitrary functions of x but the X dependence is such that Aµ (x, X) is Lie(K)-valued instead
of Lie(U (N )), i.e. eventually we have a four-dimensional gauge potential Aµ (x) with values in
Lie(K). Concerning the constraint (2.17), it is satisfied by choosing
φa = rφ(x)ωa ,
(2.21)
i.e. the unconstrained degrees of freedom correspond to the scalar field φ(x) which is a singlet
under the four-dimensional gauge group K.
The choice (2.20) defines one of the possible embedding of Lie(SU (2)) in Lie(U (N )). For
example, we could also embed Lie(SU (2)) in Lie(U (N )) using the irreducible N -dimensional
rep. of SU (2), i.e. we could identify ωa = Xa . The constraint (2.16) in this case implies that the
four-dimensional gauge group is U (1) so that Aµ (x) is U (1) valued. The constraint (2.17) leads
again to the scalar singlet φ(x).
2 . We solve the CSDR conIn general, we start with a U (1) gauge theory on M 4 × SN
straint (2.18) by embedding SU (2) in U (N ). There exist pN embeddings, where pN is the
number of ways one can partition the integer N into a set of non-increasing positive integers [24]. Then the constraint (2.16) gives the surviving four-dimensional gauge group. The
constraint (2.17) gives the surviving four-dimensional scalars and equation (2.21) is always a solution but in general not the only one. By setting φa = ωa we obtain always a minimum of the
potential. This minimum is given by the chosen embedding of SU (2) in U (N ).
Concerning fermions in the adjoint, the corresponding analysis in [19] shows that we have to
consider the embedding
S ⊂ SO(dim S),
which is given by Ta = 12 Cabc Γbc that satisfies the commutation relation (2.19). Therefore ψ
is an intertwining operator between induced representations of S in U (N P ) and in SO(dim S).
To find the surviving fermions, as in the commutative case [4], we decompose the adjoint rep.
of U (N P ) under SU (N P ) × K,
U (N P ) ⊃ SU (N P ) × K,
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
adj U (N P ) =
X
13
(si , ki ).
i
We also decompose the spinor rep. σ of SO(dim S) under S
SO(dim S) ⊃ S,
X
σ=
σe .
e
Then, when we have two identical irreps. si = σe , there is a ki multiplet of fermions surviving
in four dimensions, i.e. four-dimensional spinors ψ(x) belonging to the ki representation of K.
An important point that we would like to stress here is the question of the renormalizability
of the gauge theory defined on M4 × (S/R)F . First we notice that the theory exhibits certain
features so similar to a higher-dimensional gauge theory defined on M4 × S/R that naturally
it could be considered as a higher-dimensional theory too. For instance the isometries of the
spaces M4 × S/R and M4 × (S/R)F are the same. It does not matter if the compact space
2 , the isometries are
is fuzzy or not. For example in the case of the fuzzy sphere, i.e. M4 × SN
2
SO(3, 1) × SO(3) as in the case of the continuous space, M4 × S . Similarly the coupling of
a gauge theory defined on M4 ×S/R and on M4 ×(S/R)F are both dimensionful and have exactly
the same dimensionality. On the other hand the first theory is clearly non-renormalizable, while
the latter is renormalizable (in the sense that divergencies can be removed by a finite number
of counterterms). So from this point of view one finds a partial justification of the old hopes
for considering quantum field theories on non-commutative structures. If this observation can
lead to finite theories too, it remains as an open question.
2.4
The problem of chirality in fuzzy CSDR
Among the great successes of the ordinary CSDR is the possibility to accommodate chiral
fermions in the four-dimensional theory [7]. Needless to say that the requirement of chirality
for the four-dimensional fermions is necessary in order for a theory to have a chance to become
realistic.
Let us recall the necessary conditions for accommodating chiral fermions in four dimensions
when a higher-dimensional gauge theory with gauge group G is reduced over a d-dimensional
coset space S/R using the CSDR scheme. As we discussed previously, solving the CSDR constraints for the fermion fields leads to the result that in order to obtain the representations
of the four-dimensional unbroken gauge group H under which the four-dimensional fermions
transform, we have to decompose the representation F of the initial gauge group in which the
fermions are assigned under R × H, i.e.
X
F =
(ti , hi ),
and the spinor of the tangent space group SO(d) under R
X
σd =
σj .
Then for each pair ti and σi , where ti and σi are identical irreducible representations there is an
hi multiplet of spinor fields in the four-dimensional theory.
In order to obtain chiral fermions in four dimensions we need some further requirements. The
representation of interest, for our purposes, of the spin group is the spinor representation. This
(d−1)
d
has dimensions 2 2 and 2 2 for d even and odd respectively. For odd d the representation is
irreducible but for even d it is reducible into two irreducible components of equal dimension.
This splitting exactly gives the possibility to define Weyl spinors and to construct a chirality
14
A. Chatzistavrakidis and G. Zoupanos
operator. Thus if we are in odd number of dimensions (where the chirality operator does
not exist) there is no way to obtain chiral fermions. For this reason we focus only on even
dimensions.
The first possibility is to start with Dirac fermions in D (even) dimensions. Here we can
define the standard chirality operator
ΓD+1 = i
D(D−1)
2
Γ1 Γ2 · · · ΓD ,
with (ΓD+1 )2 = 1 and {ΓD+1 , ΓA } = 0, where ΓA , A = 1, . . . , D span the Clifford algebra in D
dimensions. This operator has eigenvalues ±1 and distinguishes left and right spinors. So, it is
possible to define a Weyl basis, where the chirality operator is diagonal, namely
ΓD+1 ψ± = ±ψ± .
As we mentioned above, in this case SO(1, D − 1) has two independent irreducible spinor rep′ , under which the Weyl spinors ψ and ψ transform respectively. The
resentations, σD and σD
+
−
following branching rule for the spinors holds4
SO(1, D − 1) ⊃ SO(1, 3) × SO(d),
σD = (2, 1; σd ) + (1, 2; σd′ ),
′
σD
= (2, 1; σd′ ) + (1, 2; σd ).
Then, since we started with a Dirac spinor ψ = ψ+ ⊕ ψ− transforming under a representation F
of the original gauge group G, following the rule which was stated above it is obvious that we
obtain fermions in four dimensions appearing in equal numbers of left and right representations
of the unbroken gauge group H. Thus, starting with Dirac fermions does not render the fermions
of the four-dimensional theory chiral.
In order to overcome this problem we can make a further restriction and start with Weyl
fermions, namely to impose the Weyl condition in higher dimensions. Then, only one of the σD
′ representations is selected. There are still two cases to investigate, the total number of
and σD
dimensions being 4n or 4n + 2. Since we are interested in vacuum configurations of the form
M 4 × S/R the dimensionality of the internal (coset) space is then of the form 4n or 4n + 2
respectively.
For D = 4n (d = 4(n − 1)), the two spinor representations of SO(d) are self-conjugate,
meaning that in the decomposition
SO(d) ⊃ R,
X
σd =
σi ,
σi is either a real representation or it appears together with its conjugate representation σ
¯i .
Thus we are led to consider that the representation F of G where the fermions are assigned
has to be complex. Two important things to note is that R is also required to admit complex
representations (otherwise the decompositions of σd and σd′ will be the same, leading to a nonchiral theory) and that rank S = rank R (otherwise σd and σd′ will again be the same). These
requirements still hold in the following case.
In the case D = 4n + 2 (d = 4(n − 1) + 2), the two spinor representations of SO(d) are not
self-conjugate anymore and σd′ = σ
¯d . Now, the decomposition reads as
SO(d) ⊃ R,
4
Here the usual notation for two-component Weyl spinors of the Lorentz group SO(1, 3) is adopted, namely
ψ+ → (1, 2) and ψ− → (2, 1).
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
σd =
σ
¯d =
X
X
15
σi ,
σ
¯i ,
so we can let F be a vectorlike representation. Then, in the decomposition
G ⊃ RG × H,
X
F =
(ti , hi ),
¯ i ). According
each term (ti , hi ) will either be self-conjugate or it will appear with the term (t¯i , h
to the established rule, σd will provide P
a left-handed fermion multiplet transforming under the
four-dimensional gaugeP
group as fL =
hL
¯d will provide a right-handed fermion multiplet
i ; σ
¯ R . Since hL ∼ h
¯ R we are led to two Weyl fermions with the same
h
transforming as fR =
i
i
i
chirality in the same representation of the unbroken gauge group H. This is of course a chiral
theory, which is the desired result. Moreover, the doubling of the fermions can be eliminated by
imposing the Majorana condition, if applicable5 .
Let us use the same spirit in order to investigate the possibility of obtaining chiral fermions
in the fuzzy case as well. We discussed previously that we have to consider the embedding
S ⊂ SO(dim S),
concerning fermions in the adjoint. In order to determine the surviving fermions, as in the
commutative case, we decompose the adjoint rep. of U (N ) under SU (N ) × K,
U (N ) ⊃ SU (N ) × K,
X
adj U (N ) =
(si , ki ).
i
We also decompose the spinor rep. σ of SO(dim S) under S
SO(dim S) ⊃ S,
X
σe .
σ=
e
Then, when we have two identical irreps. si = σe , there is a ki multiplet of fermions surviving
in four dimensions, i.e. four-dimensional spinors ψ(x) belonging in the ki representation of K.
Concerning the issue of chirality, the situation is now different. The main difference is
obviously the modification of the rule for the surviving fermions. In the continuous case we had
to embed R in SO(d), while now the suitable embedding is that of S in SO(dim S). Exploring
chirality in the continuous case, we had to deal with the representations of SO(d). Recall that
we required d to be even so that there are two independent spinor representations; therefore
in the fuzzy case we require dim S to be even. Moreover, when d = 4n we concluded that the
representation F , where the fermions are initially assigned, has to be complex. Since in the
fuzzy case we assign the fermions in the adjoint representation, the case dim S = 4n would lead
to a non-chiral theory. Finally, the case dim S = 4n + 2 is the only promising one and one would
expect to obtain chiral fermions, as in the continuous case when d = 4n + 2. However, we also
need the further requirement that S admits complex representations, again in analogy with R
admitting complex representations in the continuous case.
In summary, in order to have a chance to obtain chiral fermions in the case of fuzzy extra
dimensions the necessary requirements are:
5
Let us remind that the Majorana condition can be imposed when the number of dimensions is D = 2, 3, 8n+4.
16
A. Chatzistavrakidis and G. Zoupanos
• dim S = 4n + 2,
• S admits complex irreps.
The above requirements are quite restrictive; for example they are not satisfied in the case of
a single fuzzy sphere. In general, using elementary number theory one can show that they
cannot be satisfied for any S being a SU (n), SO(n) or Sp(n) group. Therefore only products of
fuzzy spaces have a chance to lead to chiral fermions after dimensional reduction without further
requirements. The simplest case which satisfies these requirements is that of a product of two
fuzzy spheres, which will be discussed in Section 3.4 in the context of dynamical generation of
fuzzy extra dimensions.
In conclusion it is worth making the following remark. As we saw above, a major difference
between fuzzy and ordinary CSDR is that in the fuzzy case one always embeds S in the gauge
group G instead of embedding just R in G. A generic feature of the ordinary CSDR in the
special case when S is embedded in G is that the fermions in the final theory are massive [30].
According to the discussion in Section 2.3.3 the situation in the fuzzy case is very similar to the
one we just described. In fuzzy CSDR the spontaneous symmetry breaking mechanism takes
already place by solving the fuzzy CSDR constraints. Therefore in the Yukawa sector of the
theory we have the results of the spontaneous symmetry breaking, i.e. massive fermions and
Yukawa interactions among fermions and the physical Higgs field. We shall revisit the problem
of chirality in the following section and finally, in Section 4, we shall describe a way to overcome
it and obtain chiral four-dimensional theories.
3
Dynamical generation of fuzzy extra dimensions
Let us now discuss a further development [20] of these ideas, which addresses in detail the
questions of quantization and renormalization. This leads to a slightly modified model with
an extra term in the potential, which dynamically selects a unique (nontrivial) vacuum out
of the many possible CSDR solutions, and moreover generates a magnetic flux on the fuzzy
2 .
sphere. It also allows to show that the full tower of Kaluza–Klein modes is generated on SN
Moreover, upon including fermions, the model offers the possibility of a detailed study of the
fermionic sector [21]. Such a study reveals the difficulty in obtaining chiral low-energy models
but at the same time it paves the way out of this problem. Indeed, we shall see in the following
section that using orbifold techniques it is possible to construct chiral models in the framework
of dynamically generated fuzzy extra dimensions.
3.1
The four dimensional action
We start with a SU (N ) gauge theory on four dimensional Minkowski space M 4 with coordinates y µ , µ = 0, 1, 2, 3. The action under consideration is
Z
1
†
†
SY M = d4 y T r
F
+
(D
φ
)
D
φ
F
µν
µ a
µ a − V (φ),
4g 2 µν
where Aµ are SU (N )-valued gauge fields, Dµ = ∂µ + [Aµ , ·], and
φa = −φ†a ,
a = 1, 2, 3
are three antihermitian scalars in the adjoint of SU (N ),
φa → U † φa U,
Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
17
where U = U (y) ∈ SU (N ). Furthermore, the φa transform as vectors of an additional global
SO(3) symmetry. The potential V (φ) is taken to be the most general renormalizable action
invariant under the above symmetries, which is
V (φ) = Tr (g1 φa φa φb φb + g2 φa φb φa φb − g3 εabc φa φb φc + g4 φa φa )
g5
g6
+
Tr (φa φa ) Tr (φb φb ) +
Tr(φa φb ) Tr (φa φb ) + g7 .
(3.1)
N
N
This may not look very transparent at first sight, however it can be written in a very intuitive
way. First, we make the scalars dimensionless by rescaling
φ′a = Rφa ,
where R has dimension of length; we will usually suppress R since it can immediately be reinserted, and drop the prime from now on. Now observe that for a suitable choice of R,
R=
2g2
,
g3
the potential can be rewritten as
1 †
h
2
2
˜
V (φ) = Tr a (φa φa + b1l) + c + 2 Fab Fab + gab gab
g˜
N
for suitable constants a, b, c, g˜, h, where
˜b = b + d Tr (φa φa ),
gab = Tr(φa φb ).
N
We will omit c from now. Notice that two couplings were reabsorbed in the definitions of R
and ˜b. The potential is clearly positive definite provided
Fab = [φa , φb ] − εabc φc = εabc Fc ,
a2 = g1 + g2 > 0,
2
= −g2 > 0,
g˜2
h ≥ 0,
which we assume from now on. Here ˜b = ˜b(y) is a scalar, gab = gab (y) is a symmetric tensor
under the